Find Height of a Flagpole without Approaching the PoleDate: 04/13/2006 at 10:12:02 From: Patrick Subject: find height of flagpole, with no distance to pole I am outside a fence at a right angle to a pole of unknown distance or height. I can measure along the fence and take the angles from ground level to the top of the pole from two points, exactly level with the base of the pole and 80 m apart. Points A, B, and C are exactly on level ground, A to B is 80 meters, the angle at B is a right angle. On C is a vertical pole with the top of the pole = D. Angle CBD = 32 degrees and angle CAD = 24 degrees. I am 56 yrs old and when I learned math there were no calculators. I was given this wee puzzle, found your brilliant site, and learned something about trig. Alas when this puzzle combined three dimensions with algebra and trig it was sink or swim--time I learned to swim! I can imagine two equations being equal to each other. Tan(32) = CD over BC and Tan(24) = CD over AC with CD being common so Tan(32) x BC = Tan(24) x AC. From this I know that there would be a ratio between distances AD and BD...I believe it is workable, maybe over my head but I'm interested in the answer. Thank you. Date: 04/13/2006 at 11:35:15 From: Doctor Douglas Subject: Re: find height of flagpole, with no distance to pole Hi Patrick. Good work so far! From your description I gather that angle CBA is a right angle, so that on the line that passes through A and B, the point B is in fact the closest point to the flagpole: Let h be the height of the flagpole CD. Also, let y denote the distance between B and C and Y denote the distance between A and C. The elevation angles give h = y tan(32) h = Y tan(24) The additional information that gives the relationship between the distances is the Pythagorean Theorem. Although I apply this to triangle ABC below, you can also apply it to triangle ABD, since that is a right triangle also. Because triangle ABC is a right triangle, we have Y^2 = y^2 + 80^2, and our two equations for the height h can be set equal to each other and rewritten in terms of y: y tan(32) = sqrt(y^2 + 6400) tan(24) and we can square this and solve for y: y^2 tan^2(32) = y^2 tan^2(24) + 6400 tan^2(24) y = sqrt{[6400 tan^2(24 deg)]/[tan^2(32 deg) - tan^2(24 deg)]} = sqrt(6600) = 81.24 meters And we can now calculate the height of the flagpole as h = y*tan(32 deg) = (81.24 m)*(.6249) = 50.76 meters. As a check, we also compute Y = sqrt(81.24^2 + 6400) = 114.02 meters, and (114.02 meter)*tan(24 deg) = 114.02*(.4452) = 50.76 meters, in agreement. So everything is consistent and we can be confident that our work is correct. For another approach to measuring the height of a pole without access to the base point C, see the following web page in our archives: Determine Flagpole Height without Access to the Pole http://mathforum.org/library/drmath/view/65178.html In that answer we try to keep the geometry confined to a two-dimensional plane, so that it is easier to visualize and to draw on paper. The price that we have to pay is that we have to apply trigonometry to a triangle that is not a right triangle. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ Date: 04/16/2006 at 09:15:15 From: Patrick Subject: Thank you (find height of flagpole, with no distance to pole) Very kind of you to afford your knowledge and time to satisfy my curiosity. Bless you all, I hope to be able to pass on what I have learned from you. Thanks, Patrick |
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